Beginner: need help understanding an answer

The vectors that are perpendicular to <1,1,1> and <1,2,3> lie on a ___________.

I would have said "plane".

I've worked with vectors in calculus, and if you take the cross product of those two vectors you get a vector perpendicular to both, and you could, visualizing it, move that vector around on the surface of the plane defined by <1,1,1> and <1,2,3>...

But the answer given in the solution manual is "line"...

How are all the vectors perpendicular to <1,1,1> and <1,2,3> lying on a line?

"Moving a vector around on a plane" does NOT give you a different vector! All vectors perpendicular to the given two vectors are parallel to the cross product. I'm not sure I like the wording of the problem itself! Just as moving vectors around does not give a new vector, so a vector alone does not determine a line.

and you could, visualizing it, move that vector around on the surface of the plane defined by <1,1,1> and <1,2,3>...

I don't know what you mean exactly, but if you view the vectors as arrows drawn from <0,0,0>, then the tip of the arrow you get from the cross product isn't in that plane. In fact, no vector in that plane is perpendicular to both <1,1,1> and <1,2,3>.

I don't know what you mean exactly, but if you view the vectors as arrows drawn from <0,0,0>, then the tip of the arrow you get from the cross product isn't in that plane. In fact, no vector in that plane is perpendicular to both <1,1,1> and <1,2,3>.